Is there any "deep" relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:22:19Z http://mathoverflow.net/feeds/question/106830 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106830/is-there-any-deep-relation-between-the-localization-theorem-of-equivariant-coho Is there any "deep" relation between the localization theorem of equivariant cohomology and the localization theorem of equivariant K-theory Zhaoting Wei 2012-09-10T16:22:43Z 2012-09-10T16:22:43Z <p>First let's consider equivariant cohomology: if a compact Lie group $G$ acts on a compact manifold $M$. We have the equivariant cohomology $H_G(M)$ defined as the cohomology of the cochain complex $((\mathbb{C}[\mathfrak{g^*}]\otimes \Omega^{\bullet}{M})^G, d_G)$ (for the definition of equivariant cohomology we can look at chapter 1 and 4 of Guillemin and Sternberg's book "Supersymmetry and Equivariant de Rham Theory"). $K&lt; G$ is a closed subgroup, Let $M^K$ be the points of $M$ which has isotropy groups conjugated to $K$, obviously $M^K$ is a $G$-submanifold of $M$ and let $~i: M^K \rightarrow M$ denote the inclusion map. we have a version of localization theorem, see Guillemin and Sternberg's book "Supersymmetry and Equivariant de Rham Theory" chapter 11, especially Theorem 11.4.3 in page 178. In more details :</p> <p>Consider the equivariant cohomology $H_G(M)$ and $H_G(M^K)$ as $S( \mathfrak{g^* })^G$ modules. Then the pullback map $$i^*: H^ * _G(M)\rightarrow H^ *_G(M^K)$$ is an isomorphism after localizing at some certain prime ideals of $S( \mathfrak{g^* })^G$.</p> <p>On the other hand, we have the equivariant K-theory $K_G(M)$ and we also have the localization theorem in this side, see Segal "Equivariant K-theory" (1967) section 4, proposition 4.1, which also claims that Then the pullback map $$i^*: K^ * _G(M)\rightarrow K^ *_G(M^K)$$ is an isomorphism after localizing at some certain prime ideals of $R(G)$, the representation ring of $G$.</p> <p>We notice the similarity of the above two version of localization theorems. Nevertheless equivariant cohomology and equivariant K-theory are different. The first is the cohomology of a differential graded algebra and the latter is the Grothedieck group of modules of the cross product algebra $G \ltimes C(M)$.</p> <p>My question is: is there any deep relation between them? Are they valid because of the same reason?</p>